Inegalitate conditionata 3

Marius Mainea · Post by **Marius Mainea** » Sat Dec 20, 2008 11:30 pm

Fie a,b,c nenegative cu a+b+c=1. Demonstrati ca:

\( \frac{a^2b^2}{1-ab}+\frac{b^2c^2}{1-bc}+\frac{c^2a^2}{1-ca}\le \frac{1}{12}. \)

Marius Mainea · Post by **Marius Mainea** » Mon Jan 05, 2009 11:04 pm

Avem \( ab\le(\frac{a+b}{2})^2\le (\frac{a+b+c}{2})^2=\frac{1}{4} \) si analoagele deci

\( LHS\le\frac{4}{3}(a^2b^2+b^2c^2+c^2a^2)\le \frac{1}{12} \)(demonstrati)

Marius Mainea · Post by **Marius Mainea** » Wed Jan 07, 2009 9:01 pm

Marius Mainea wrote:
\( LHS\le\frac{4}{3}(a^2b^2+b^2c^2+c^2a^2)\le \frac{1}{12} \)(demonstrati)

Fie a= max(a,b,c)

\( ab+ac=a(b+c)\le (\frac{a+b+c}{2})^2=\frac{1}{4} \)